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Theorem ntrneifv3 38906
Description: The value of the neighbors (convergents) expressed in terms of the interior (closure) function. (Contributed by RP, 26-Jun-2021.)
Hypotheses
Ref Expression
ntrnei.o 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
ntrnei.f 𝐹 = (𝒫 𝐵𝑂𝐵)
ntrnei.r (𝜑𝐼𝐹𝑁)
ntrnei.x (𝜑𝑋𝐵)
Assertion
Ref Expression
ntrneifv3 (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠)})
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠   𝑘,𝐼,𝑙,𝑚   𝑁,𝑠   𝑋,𝑙,𝑚,𝑠   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠
Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝐼(𝑖,𝑗,𝑠)   𝑁(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝑋(𝑖,𝑗,𝑘)

Proof of Theorem ntrneifv3
StepHypRef Expression
1 dfin5 3731 . 2 (𝒫 𝐵 ∩ (𝑁𝑋)) = {𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)}
2 ntrnei.o . . . . . . 7 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
3 ntrnei.f . . . . . . 7 𝐹 = (𝒫 𝐵𝑂𝐵)
4 ntrnei.r . . . . . . 7 (𝜑𝐼𝐹𝑁)
52, 3, 4ntrneinex 38901 . . . . . 6 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))
6 elmapi 8031 . . . . . 6 (𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
75, 6syl 17 . . . . 5 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
8 ntrnei.x . . . . 5 (𝜑𝑋𝐵)
97, 8ffvelrnd 6503 . . . 4 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
109elpwid 4309 . . 3 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
11 sseqin2 3968 . . 3 ((𝑁𝑋) ⊆ 𝒫 𝐵 ↔ (𝒫 𝐵 ∩ (𝑁𝑋)) = (𝑁𝑋))
1210, 11sylib 208 . 2 (𝜑 → (𝒫 𝐵 ∩ (𝑁𝑋)) = (𝑁𝑋))
134adantr 466 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
148adantr 466 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
15 simpr 471 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
162, 3, 13, 14, 15ntrneiel 38905 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑋)))
1716bicomd 213 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑠 ∈ (𝑁𝑋) ↔ 𝑋 ∈ (𝐼𝑠)))
1817rabbidva 3338 . 2 (𝜑 → {𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)} = {𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠)})
191, 12, 183eqtr3a 2829 1 (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351  cin 3722  wss 3723  𝒫 cpw 4297   class class class wbr 4786  cmpt 4863  wf 6027  cfv 6031  (class class class)co 6793  cmpt2 6795  𝑚 cmap 8009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-map 8011
This theorem is referenced by: (None)
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